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arXiv · 1309.2877

Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

Fredrik Johansson
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Abstract

We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function $\zeta(s,a)$ for $s, a \in \mathbb{C}$, along with an arbitrary number of derivatives with respect to $s$, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.

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Fredrik Johansson. 2013-09-11. Rigorous high-precision computation of the Hurwitz zeta function and its derivatives. https://arxiv.org/abs/1309.2877

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